Model-Based Reinforcement Learning Explore a comprehensive and practical approach to reinforcement learning
Reinforcement learning is an essential paradigm of machine learning, wherein an intelligent agent performs actions that ensure optimal behavior from devices. While this paradigm of machine learning has gained tremendous success and popularity in recent years, previous scholarship has focused either on theory—optimal control and dynamic programming – or on algorithms—most of which are simulation-based.
Model-Based Reinforcement Learning provides a model-based framework to bridge these two aspects, thereby creating a holistic treatment of the topic of model-based online learning control. In doing so, the authors seek to develop a model-based framework for data-driven control that bridges the topics of systems identification from data, model-based reinforcement learning, and optimal control, as well as the applications of each. This new technique for assessing classical results will allow for a more efficient reinforcement learning system. At its heart, this book is focused on providing an end-to-end framework—from design to application—of a more tractable model-based reinforcement learning technique.
Model-Based Reinforcement Learning readers will also find:
A useful textbook to use in graduate courses on data-driven and learning-based control that emphasizes modeling and control of dynamical systems from data
Detailed comparisons of the impact of different techniques, such as basic linear quadratic controller, learning-based model predictive control, model-free reinforcement learning, and structured online learning
Applications and case studies on ground vehicles with nonholonomic dynamics and another on quadrator helicopters
An online, Python-based toolbox that accompanies the contents covered in the book, as well as the necessary code and data
Model-Based Reinforcement Learning is a useful reference for senior undergraduate students, graduate students, research assistants, professors, process control engineers, and roboticists.
About the Authors xi Preface xiii Acronyms xv Introduction xvii 1 Nonlinear Systems Analysis 1 1.1 Notation 1 1.2 Nonlinear Dynamical Systems 2 1.2.1 Remarks on Existence, Uniqueness, and Continuation of Solutions 2 1.3 Lyapunov Analysis of Stability 3 1.4 Stability Analysis of Discrete Time Dynamical Systems 7 1.5 Summary 10 Bibliography 10 2 Optimal Control 11 2.1 Problem Formulation 11 2.2 Dynamic Programming 12 2.2.1 Principle of Optimality 12 2.2.2 Hamilton–Jacobi–Bellman Equation 14 2.2.3 A Sufficient Condition for Optimality 15 2.2.4 Infinite-Horizon Problems 16 2.3 Linear Quadratic Regulator 18 2.3.1 Differential Riccati Equation 18 2.3.2 Algebraic Riccati Equation 23 2.3.3 Convergence of Solutions to the Differential Riccati Equation 26 2.3.4 Forward Propagation of the Differential Riccati Equation for Linear Quadratic Regulator 28 2.4 Summary 30 Bibliography 30 3 Reinforcement Learning 33 3.1 Control-Affine Systems with Quadratic Costs 33 3.2 Exact Policy Iteration 35 3.2.1 Linear Quadratic Regulator 39 3.3 Policy Iteration with Unknown Dynamics and Function Approximations 41 3.3.1 Linear Quadratic Regulator with Unknown Dynamics 46 3.4 Summary 47 Bibliography 48 4 Learning of Dynamic Models 51 4.1 Introduction 51 4.1.1 Autonomous Systems 51 4.1.2 Control Systems 51 4.2 Model Selection 52 4.2.1 Gray-Box vs. Black-Box 52 4.2.2 Parametric vs. Nonparametric 52 4.3 Parametric Model 54 4.3.1 Model in Terms of Bases 54 4.3.2 Data Collection 55 4.3.3 Learning of Control Systems 55 4.4 Parametric Learning Algorithms 56 4.4.1 Least Squares 56 4.4.2 Recursive Least Squares 57 4.4.3 Gradient Descent 59 4.4.4 Sparse Regression 60 4.5 Persistence of Excitation 60 4.6 Python Toolbox 61 4.6.1 Configurations 62 4.6.2 Model Update 62 4.6.3 Model Validation 63 4.7 Comparison Results 64 4.7.1 Convergence of Parameters 65 4.7.2 Error Analysis 67 4.7.3 Runtime Results 69 4.8 Summary 73 Bibliography 75 5 Structured Online Learning-Based Control of Continuous-Time Nonlinear Systems 77 5.1 Introduction 77 5.2 A Structured Approximate Optimal Control Framework 77 5.3 Local Stability and Optimality Analysis 81 5.3.1 Linear Quadratic Regulator 81 5.3.2 SOL Control 82 5.4 SOL Algorithm 83 5.4.1 ODE Solver and Control Update 84 5.4.2 Identified Model Update 85 5.4.3 Database Update 85 5.4.4 Limitations and Implementation Considerations 86 5.4.5 Asymptotic Convergence with Approximate Dynamics 87 5.5 Simulation Results 87 5.5.1 Systems Identifiable in Terms of a Given Set of Bases 88 5.5.2 Systems to Be Approximated by a Given Set of Bases 91 5.5.3 Comparison Results 98 5.6 Summary 99 Bibliography 99 6 A Structured Online Learning Approach to Nonlinear Tracking with Unknown Dynamics 103 6.1 Introduction 103 6.2 A Structured Online Learning for Tracking Control 104 6.2.1 Stability and Optimality in the Linear Case 108 6.3 Learning-based Tracking Control Using SOL 111 6.4 Simulation Results 112 6.4.1 Tracking Control of the Pendulum 113 6.4.2 Synchronization of Chaotic Lorenz System 114 6.5 Summary 115 Bibliography 118 7 Piecewise Learning and Control with Stability Guarantees 121 7.1 Introduction 121 7.2 Problem Formulation 122 7.3 The Piecewise Learning and Control Framework 122 7.3.1 System Identification 123 7.3.2 Database 124 7.3.3 Feedback Control 125 7.4 Analysis of Uncertainty Bounds 125 7.4.1 Quadratic Programs for Bounding Errors 126 7.5 Stability Verification for Piecewise-Affine Learning and Control 129 7.5.1 Piecewise Affine Models 129 7.5.2 MIQP-based Stability Verification of PWA Systems 130 7.5.3 Convergence of ACCPM 133 7.6 Numerical Results 134 7.6.1 Pendulum System 134 7.6.2 Dynamic Vehicle System with Skidding 138 7.6.3 Comparison of Runtime Results 140 7.7 Summary 142 Bibliography 143 8 An Application to Solar Photovoltaic Systems 147 8.1 Introduction 147 8.2 Problem Statement 150 8.2.1 PV Array Model 151 8.2.2 DC-D C Boost Converter 152 8.3 Optimal Control of PV Array 154 8.3.1 Maximum Power Point Tracking Control 156 8.3.2 Reference Voltage Tracking Control 162 8.3.3 Piecewise Learning Control 164 8.4 Application Considerations 165 8.4.1 Partial Derivative Approximation Procedure 165 8.4.2 Partial Shading Effect 167 8.5 Simulation Results 170 8.5.1 Model and Control Verification 173 8.5.2 Comparative Results 174 8.5.3 Model-Free Approach Results 176 8.5.4 Piecewise Learning Results 178 8.5.5 Partial Shading Results 179 8.6 Summary 182 Bibliography 182 9 An Application to Low-level Control of Quadrotors 187 9.1 Introduction 187 9.2 Quadrotor Model 189 9.3 Structured Online Learning with RLS Identifier on Quadrotor 190 9.3.1 Learning Procedure 191 9.3.2 Asymptotic Convergence with Uncertain Dynamics 195 9.3.3 Computational Properties 195 9.4 Numerical Results 197 9.5 Summary 201 Bibliography 201 10 Python Toolbox 205 10.1 Overview 205 10.2 User Inputs 205 10.2.1 Process 206 10.2.2 Objective 207 10.3 SOL 207 10.3.1 Model Update 208 10.3.2 Database 208 10.3.3 Library 210 10.3.4 Control 210 10.4 Display and Outputs 211 10.4.1 Graphs and Printouts 213 10.4.2 3D Simulation 213 10.5 Summary 214 Bibliography 214 A Appendix 215 A.1 Supplementary Analysis of Remark 5.4 215 A.2 Supplementary Analysis of Remark 5.5 222 Index 223
Milad Farsi received the B.S. degree in Electrical Engineering (Electronics) from the University of Tabriz in 2010. He obtained his M.S. degree also in Electrical Engineering (Control Systems) from the Sahand University of Technology in 2013. Moreover, he gained industrial experience as a Control System Engineer between 2012 and 2016. Later, he acquired the Ph.D. degree in Applied Mathematics from the University of Waterloo, Canada, in 2022, and he is currently a Postdoctoral Fellow at the same institution. His research interests include control systems, reinforcement learning, and their applications in robotics and power electronics. Jun Liu received the Ph.D. degree in Applied Mathematics from the University of Waterloo, Canada, in 2010. He is currently an Associate Professor of Applied Mathematics and a Canada Research Chair in Hybrid Systems and Control at the University of Waterloo, Canada, where he directs the Hybrid Systems Laboratory. From 2012 to 2015, he was a Lecturer in Control and Systems Engineering at the University of Sheffield. During 2011 and 2012, he was a Postdoctoral Scholar in Control and Dynamical Systems at the California Institute of Technology. His main research interests are in the theory and applications of hybrid systems and control, including rigorous computational methods for control design with applications in cyber-physical systems and robotics.
